Smoothing functions, cubic spline

A convenient and systematic way to represent fj (rtj) (r is the distance between particles i and j) as a linear function of unknowns is to employ cubic splines [48], as shown in Figure 8-3. The advantage of using cubic splines is that the function is continuous not only across the mesh points, but also in the first and second derivatives. This ensures a smooth curvature across the mesh points. The distance is divided into 1-dimensional mesh points, thus, fj rij) in the Mi mesh (r < rq < r +i) is described by Eqs. (8-4), (8-5) and (8-6) [48],... 

In the present study we have extracted the EXAFS from the experimentally recorded X-ray absorption spectra following the method described in detail in Ref. (l , 20). In this procedure, a value for the energy threshold of the absorption edge is chosen to convert the energy scale into k-space. Then a smooth background described by a set of cubic splines is subtracted from the EXAFS in order to separate the non-osciHatory part in ln(l /i) and, finally, the EXAFS is multiplied by a factor k and divided by a function characteristic of the atomic absorption cross section (20). 

In addition, we are interested in functions that are at least twice continuously differentiable. One can draw several such curves satisfying (4.27), and the "smoothest" of them is the one minimizing the integral (4.19). It can be shown that the solution of this constrained minimization problem is a natural cubic spline (ref. 12). We call it smoothing spline. 

Spline Fitting in One Variable. By definition, a cubic spline function in one variable consists of a set of polynomial arcs of degree three or less joined smoothly end to end. The smoothness consists of continuity in the function itself and in its first and second derivatives. 

A common method of extracting f K) from Eq. 3.82 is to assume a form of the distribution function by differentiation of a smooth fimction describing the data. The function obtained by this method is called the affinity spectrum (AS) and the method, the AS method [71]. The most general approach uses a cubic spline to approximate the data. However, a simpler procedure uses a Langmuir-Freundlich (LF) isotherm model and the AS distribution is derived from the best parameters of a fit of the experimental isotherm data to the LF model [71]. This approach yields a unimodal distribution of binding affinity with a central peak, if the range... 

Make a cubic B-spline pass through all the data points [x(z),))(/), i = 1,..., zm], A cubic spline is a cubic function of position, defined on small regions between data points. It is constructed so the function and its first and second derivatives are continuous from one region to another. It usually makes a nice smooth curve through the points. The following commands create Figure B.3. 

Another structure for expressing a nonlinear relationship between X and Y is splines [333] or smoothing functions [75]. Splines are piecewise polynomials joined at knots (denoted by Zj) with continuity constraints on the function and all its derivatives except the highest. Splines have good approximation power, high flexibility and smooth appearance as a result of continuity constraints. For example, if cubic splines are used for representing the inner relation ... 

Fig. 7. Projection plot of the X-ray scattering from a solution of oscillating microtubules, showing the X-ray intensity (S I(S), z-axis) as a function of scattering vector (S = 2 sin 9/lambda, x-axis) and time (y-axis, 3 sec scan interval). Microtubule protein, 32 mg/ml. The central scatter (left) indicates overall assembly, the subsidiary maximum arises from microtubules. The temperature jump is at time zero. The periodicity of the fluctuations is about 2 min. The final state (after disappearance of the oscillations) is dominated by the scattering from oligomers. The scattering curves here and in Fig. 8 have been smoothed by cubic splines. From [16]...

The approach used by Fisher et al. (1995) is a smoothed cubic spline that approximates the forward ciuve. The number of nodes to use is recommended as approximately one-third of the number of bonds used in the sample, spaced apart so that there is an equal number of bonds maturing between adjacent nodes. This is different to the theoretical approach, which is to have node points at every interval where there is a bond cash flow however, in practice using the smaller number of nodes as proposed by Fisher et al. produces essentially an identical forward rate curve, but with fewer calculations required. The resulting forward rate curve is the cubic spline that minimises the function (5.17) ... 

The x-axis in the regression is divided into segments at the knot points, at each of which the slopes of adjoining curves on either side of the point must match, as must the curvatures. FIGURE 5.4 shows a cubic spline with knot points at 0, 2, 5, 10, and 25 years, at each of which the curve is a cubic polynomial. This function permits a high and low to be accommodated in each space bounded by the knot points. The values of the curve can be adjoined at the knot point in a smooth function. 

To approximate experimental adsorption isotherms and to calculate appropriate derivatives, we apply a procedure of smoothing splines, described by Reinsch [10]. In this approach, the experimental data are approximated by a cubic spline function g(x), which minimizes the following functional... 

An iterative solution of Eq. 15 then yields Eb (%iax)- H proved difficult to accurately fit Eq1 (9) to a polynomial in ic, however an accurate (smoothed) fit was possible using cubic splines. This fitted barrier function will be denoted Eq1 s (9)... 

The heat capacities of LUCI3 measured experimentally and smoothed by a cubic spline function were reported by Tolmach et al. (1987c). We calculated 6r>, 6 1, 0e2, 0e3, and a on the basis of all these experimental data, except three values in the temperature range 254.24-262.41 K. The characteristic parameters listed in Table 22 were obtained at a comparatively low = 0.05497 value. Comparison with the corresponding data on hexagonal lanthanide trichlorides shows that an increase in the molar volume causes an insignificant decrease in a, noticeable increases in 0 2 and 0E3, and substantial decreases in 0 and 0ei- Such modifications of these parameters change the temperature dependence of heat capacity in a quite definite way. Namely, because of the low 0 and 6ei values, heat... 

Equations (4) and (5) are not evaluated explicitly in the minimization program, but are fit using a combination of spline [17] methods, which provide stability, the ability to filter noise easily, and the flexibility to describe an arbitrarily shaped potential curve. Moreover, the final functional form is inexpensive to evaluate, making it amenable to global minimization. The initial step in our methodology is to fit the statistical pair data for each amino acid and for the density profile to Bezier splines [17]. In contrast to local representations such as cubic splines, the Bezier spline imposes global as well as local smoothness and hence effectively eliminates the random oscillatory behavior observed in our data. 

In mathematics a spline is a piecewise polynomial function, made up of individual polynomial sections or segments that are joined together at (user-selected) points known as knot points. Splines used in term structure modeling are generally made up of cubic polynomials. The reason they are often cubic polynomials, as opposed to polynomials of order, say, two or five, is explained in straightforward fashion by de la Grandville (2001). A cubic spline is a function of order three and a piecewise cubic polynomial that is twice differentiable at each knot point. At each knot point the slope and curvature of the curve on either side must match. The cubic spline approach is employed to fit a smooth curve to bond prices (yields) given by the term discount factors. 

This section is concerned with outlining the procedures used for data analysis, after the collection of intensity data. We start with an intensity function /g p (x) which has been obtained by step scanning at intervals of 0.2 20 (or preferably in equal intervals of x) from say, 3 to 110°. Such a data function has an element of noise arising from the statistical nature of x-ray production, scattering, and recording. This noise may be minimized by the use of an appropriate smoothing function. We have found the procedures using cubic splines described by Dixon et al. particularly suitable for this purpose. The use of cubic splines has the utility that the data function may be easily... 

---